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A Bayesian approach to inferring vascular tree structure from 2D imagery
by
E. Thönnes, A. Bhalerao, W.S. Kendall and R.G. Wilson
October 2001
DEPARTMENT OF STATISTICS
UNIVERSITY OF WARWICK
A BAYESIAN APPROACH TO INFERRING VASCULAR TREE STRUCTURE FROM
2D IMAGERY
Elke Thönnes , Abhir Bhalerao , Wilfrid Kendall , and Roland Wilson
Departments of Computer Science and Statistics
University of Warwick, UK
elke|wsk @stats.warwick.ac.uk
abhir|rgw @dcs.warwick.ac.uk
ABSTRACT
We describe a method for inferring tree-like vascular structures from 2D imagery. A Markov Chain Monte Carlo
(MCMC) algorithm is employed to sample from the posterior distribution given local feature estimates, derived from
likelihood maximisation for a Gaussian intensity profile. A
multiresolution scheme, in which coarse scale estimates are
used to initialise the algorithm for finer scales, has been implemented and used to model retinal images. Results are
presented to show the effectiveness of the method.
1. INTRODUCTION
The problem of inferring vascular structure from image data
is an important one, especially in the area of surgical planning, which requires both efficient computation and effective use of prior knowledge. Previous work in the area has
tended to focus on the modelling of specific vascular features [1] or to use approaches such as adaptive thresholding
[4].
The aim of the work described here is to formulate a
general method for the inference, which can be applied in
two or three dimensions and makes effective use of prior
knowledge, yet which is sufficiently general to be applied
to a wide range of problems. The common statistical methods for such medical image analysis have typically used
likelihood techniques, such as Expectation-Maximisation
(EM) [6, 5]. Although EM methods can be efficient computationally, they have only limited scope for incorporating prior knowledge. A more powerful way of including
prior information is to use a Bayesian method, such as maximum a posteriori (MAP) estimation. The principal difficulty with Bayesian techniques is computational: they normally require the use of Markov chain Monte Carlo algorithms, which may run for hundreds of thousands of iterations to yield reliable results [3]. This has restricted their
This work was supported by the UK EPSRC.
use in applications involving large data sets, such as medical
images.
The method we have adopted combines local likelihood
maximisation using a Gaussian model of the spatial intensity profile, on a multiresolution grid and global structure
determination using a Bayesian technique derived from a
general model of vasculature as a collection of tree structures. By approaching the problem in this way, we combine
the computational efficiency of likelihood techniques with
the power and generality of a Bayesian approach. After a
brief description of the estimation algorithms, we present
results of two dimensional structure inference on real data.
The paper is concluded with some observations on the technique.
2. A STATISTICAL MODEL OF VASCULAR
STRUCTURE
To draw inferences about global structure, we employ a
Bayesian formalism: the data are modelled as a random
tree-like structure and we then use an MCMC algorithm [2]
to sample from the posterior distribution, which is conditioned on the data. The sampling distribution is an approximate equilibrium of a random process whose configuration
space is the space of tree-like structures and whose equilibrium is designed to be the target conditional distribution.
As well as gaining information about the global structure,
variation in the posterior samples enables us to quantify uncertainties about the image interpretation.
The prior distribution defines the global structure as a forest of a random number of trees. Each such tree is binary:
branches divide only into two sub-branches at any given
position. A physical realisation of such a tree needs each
vertex to be located in space. Unfortunately, the simplistic approach of displacing each vertex from its parent by
a Gaussian displacement of zero mean (a “random-walk”
tree) leads to a tangled local structure (left hand figure 1).
We therefore introduce a correlation by allowing the mean
displacement to be a small linear multiple of the displacement of the parent vertex from the grandparent vertex (an
“AR(1)” tree) (right hand of figure 1). To model intensity,
with each vertex in the tree we associate a Gaussian kernel that represents the spatial grey level profile of the corresponding vessel segment.
of
The posterior distribution for a random number
trees is given by
+*$, . - - / 02123 5 4 6 "!$#&%(')
78 9;:<=
Fig. 1. Illustration of a random-walk tree and an AR(1) tree
(1)
>?
,
where is the image. The distribution of , penalises the number of trees; in the examples, a Poisson
distribution was used. This ensures that a ‘minimal’ expla are the probabilities
nation of the data is found.
%A@ %
%(B process and C D4 EGF
defining the degree of the branching
is the degree of the vertex 4 . , H- - / 0I1J3 54 is the
distribution of the parameters of the vertex, - . In the examples, this is an autoregressive (AR(1)) process. The parameters - represent the positions of vertices, K , and the
amplitude and width parameters of the edges, while 8 L:&
is the likelihood function. The observation model is based
on the approximation of linear structures by a sum of Gaussian kernels:
NM
=Q
;
K 9"!$#PO
KSR
K parent +* ETK &V U)
(2)
where is a magnitude scale factor and K =V the mean
O
and covariance
parameters of the kernel. Rather than sample
directly on the covariance parameters, they are sampled in
a normalised form, in terms of the orientation and the standard deviations of their principal components. The orientation is defined by the line joining a vertex with its parent.
The likelihood is then computed based on a normal model
of the observations within a region
K;
M
A
K WEYX KA
graft) and the chances of each move are balanced against
its opposite, it is straightforward to compute the required
equilibrium distribution and to design the move probabilities to give the required posterior as equilibrium, using the
Metropolis-Hastings technique (MH) [3]. The MH method
iterates in two steps: the first proposes a move from the collection of moves and the second accepts or rejects the move
so as to ensure that the equilibrium coincides with the desired posterior. The randomised decision whether to accept
or reject a move depends on whether the resulting new tree
will be a more adequate representation from the posterior
than the current tree.
The efficiency of the algorithm depends crucially on the
proposed moves. As all moves influence the likelihood only
locally, the likelihood evaluation can be implemented efficiently. Global structure that can be inferred with high
certainty locally will lead to a tree structure that is stable
over time, while low local certainty results in a volatile treestructure that alternates between different explanations for
the global structure. A summary of the moves currently implemented for the algorithm is shown in figure 2.
Each iteration of the sampler consists of the random selection of one such move and its random acceptance or rejection, according to posterior probability.
3. EXPERIMENTS
(3)
where X K; respresents the noise in the observations, which
again is assumed to be normally distributed, with zero mean
and independent at each pixel. The simulation, whose configuration at any one time is a collection of random trees,
is designed to have moves which take it from one configuration to the next. In order to arrive at a Maximum a Posteriori
(MAP) estimate from a given image, it is necessary to sample from the above posterior distribution. This is done by
a simulation involving a number of different moves, affecting the state of the forest at each iteration. As long as each
move has an ‘opposite’ (e.g. add versus delete, split versus
Figure 3 illustrates results of the ML model estimation. We
have used part of a 2D retinal angiographic image size 404
7
404 pixels (fig. 3(a)) for our 2D experiments. To improve the efficiency of the estimation, we start from a set of
maximum likelihood local feature estimates, based on the
Gaussian kernel of (3), within a multiresolution framework,
combined with a piecewise linear model of the background
intensity to account for the obvious large scale variation in
intensity across the images such as fig. 3(a). In other words,
each region in a quadtree tessellation of the image is modelled as the sum of a linear background term and a Gaussian
intensity profile, whose parameters are estimated using an
structure. After the first 200 iterations, the scale is halved
and the appropriate local feature estimates are used to guide
the sampler; after 3000, the scale is halved again, as it is
after 6000 iterations, at which point, the highest spatial resolution is reached. This approach has been found to speed
convergence to the equilibrium distribution, while avoiding
becoming trapped in local modes, in a similar manner to
many coarse-fine algorithms. Note that one iteration consists of the generation and acceptance of a single proposal
(for ‘editing’ the tree). In other words, 100000 iterations
is comparable, in terms of computation, to a single scan
through the image. It has been noted from experiments that
equilibrium is reached in approximately 50000 iterations, a
comparatively low burden computationally.
4. CONCLUSIONS
Fig. 2. Moves used in MCMC simulation on trees.
iterative, EM-type algorithm to maximise likelihood
O
M
KA Q
K2R
K &V M
K R
B S
Q
K =V (4)
K M
V U M
M
KA
KA Q
K R
5. REFERENCES
K; Q
K R
K &V KSR
K K R
K KA Q
M
K; Q
K R
K Some encouraging preliminary results have been achieved
using the approach described in section 2, demonstrating
its potential for modelling vascular structure globally in a
computationally efficient way. Fine-tuning the algorithm
will lead to significant improvements. These will include,
for example, the use of the local estimates to produce initial
configurations for the MCMC algorithm. Such improvements are currently being implemented. The work is also
being tested on other types of data and extended to three
dimensions.
K &V (5)
K2R
K &V (6)
where the data K; are windowed with a cosine window,
whose size is twice the block width at a given scale, to
give the data KA used in the estimator. The index denotes iteration number; typically 4-5 iterations are sufficient to give accurate estimates. Figures 3(b)-(c) show
reconstructions using the 2D Gaussians in each block (at
corresponding block sizes) based on the ML feature estimates at block sizes of 7 and F 7 F respectively.
Clearly, at lower spatial resolutions, the model cannot easily describe the presence of multiple vessels within the window, such as occur at bifurcations, and the resulting lowamplitude, isotropic Gaussians are locally the ‘best’ description of these regions. However, these blocks can be
modelled accurately at higher spatial resolutions. The second set of images shows how the local estimates from different window scales is used in a coarse-fine stochastic simulation, in order to get a Bayesian estimate of the forest
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[4] M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton,
A. S. Thom, A. A. Bharath, and K. H. Parker. Segmentation of retinal blood vessels based on the second
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[5] W. M. Wells, R. Kikinis, W. E. L. Grimson, and
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(a)
(a)
(b)
(b)
(c)
(c)
Fig. 3. (a) 2D retinal angiogram size 404 7 404 pixels.
(b)
model parameters estimates
Reconstruction
for block sizesof ofdata64from
and (c) 16.
Fig. 4. Estimates from the tree-based sampler, at (a) 200, (b)
1000 and (c) 50000 iterations, showing how use is made of
the local multiresolution feature estimates, in a coarse-fine
approach to the MCMC algorithm. After 50000 iterations,
few changes occur.
Other University of Warwick Department of Statistics
Research Reports authored or co–authored by W.S. Kendall.
161: The Euclidean diffusion of shape.
162: Probability, convexity, and harmonic maps with small image I: Uniqueness and fine existence.
172: A spatial Markov property for nearest–neighbour Markov point processes.
181: Convexity and the hemisphere.
202: A remark on the proof of Itô’s formula for C 2 functions of continuous
semimartingales.
203: Computer algebra and stochastic calculus.
212: Convex geometry and nonconfluent Γ-martingales I: Tightness and
strict convexity.
213: The Propeller: a counterexample to a conjectured criterion for the
existence of certain convex functions.
214: Convex Geometry and nonconfluent Γ-martingales II: Well–posedness
and Γ-martingale convergence.
216: (with E. Hsu) Limiting angle of Brownian motion in certain two–
dimensional Cartan–Hadamard manifolds.
217: Symbolic Itô calculus: an introduction.
218: (with H. Huang) Correction note to “Martingales on manifolds and
harmonic maps.”
222: (with O.E. Barndorff-Nielsen and P.E. Jupp) Stochastic calculus, statistical asymptotics, Taylor strings and phyla.
223: Symbolic Itô calculus: an overview.
231: The radial part of a Γ-martingale and a non-implosion theorem.
236: Computer algebra in probability and statistics.
237: Computer algebra and yoke geometry I: When is an expression a tensor?
238: Itovsn3: doing stochastic calculus with Mathematica.
239: On the empty cells of Poisson histograms.
244: (with M. Cranston and P. March) The radial part of Brownian motion
II: Its life and times on the cut locus.
247: Brownian motion and computer algebra (Text of talk to BAAS Science
Festival ’92, Southampton Wednesday 26 August 1992, with screenshots of illustrative animations).
257: Brownian motion and partial differential equations: from the heat equation to harmonic maps (Special invited lecture, 49th session of the ISI,
Firenze).
260: Probability, convexity, and harmonic maps II: Smoothness via probabilistic gradient inequalities.
261: (with G. Ben Arous and M. Cranston) Coupling constructions for hypoelliptic diffusions: Two examples.
280: (with M. Cranston and Yu. Kifer) Gromov’s hyperbolicity and Picard’s
little theorem for harmonic maps.
292: Perfect Simulation for the Area-Interaction Point Process.
293: (with A.J. Baddeley and M.N.M. van Lieshout) Quermass-interaction
processes.
295: On some weighted Boolean models.
296: A diffusion model for Bookstein triangle shape.
301: COMPUTER ALGEBRA: an encyclopaedia article.
308: Perfect Simulation for Spatial Point Processes.
319: Geometry, statistics, and shape.
321: From Stochastic Parallel Transport to Harmonic Maps.
323: (with E. Thönnes) Perfect Simulation in Stochastic Geometry.
325: (with J.M. Corcuera) Riemannian barycentres and geodesic convexity.
327: Symbolic Itô calculus in AXIOM: an ongoing story.
328: Itovsn3 in AXIOM: modules, algebras and stochastic differentials.
331: (with K. Burdzy) Efficient Markovian couplings: examples and counterexamples.
333: Stochastic calculus in Mathematica: software and examples.
341: Stationary countable dense random sets.
347: (with J. Møller) Perfect Metropolis-Hastings simulation of locally stable
point processes.
348: (with J. Møller) Perfect implementation of a Metropolis-Hastings simulation of Markov point processes
349: (with Y. Cai) Perfect simulation for correlated Poisson random variables conditioned to be positive.
350: (with Y. Cai) Perfect implementation of simulation for conditioned
Boolean Model via correlated Poisson random variables.
353: (with C.J. Price) Zeros of Brownian Polynomials.
371: (with G. Montana) Small sets and Markov transition densities.
382: (with A. Brix) Simulation of cluster point processes without edge effects.
391: (with E. Thönnes, A. Bhalerao, R.G. Wilson) A Bayesian approach to
inferring vascular tree structure from 2D imagery.
392: (with A. Bhalerao, E. Thönnes, R.G. Wilson) Inferring vascular tree
structure from 2D and 3D imagery.
Also see the following related preprints
317: E. Thönnes: Perfect Simulation of some point processes for the impatient user.
334: M.N.M. van Lieshout and E. Thönnes: A Comparative Study on the
Power of van Lieshout and Baddeley’s J-function.
359: E. Thönnes: A Primer on Perfect Simulation.
366: J. Lund and E. Thönnes: Perfect Simulation for point processes given
noisy observations.
If you want copies of any of these reports then please email your requests to
the secretary using statistics@warwick.ac.uk (mail address: the Department of Statistics, University of Warwick, Coventry CV4 7AL, UK).